03 knowledge representations

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Knowledge Representations : AI Course Lecture 15 22, notes, slides

www.myreaders.info/ , RC Chakraborty, e-mail [email protected] , June 01, 2010

www.myreaders.info/html/artificial_intelligence.html

Knowledge Representation

Issues, Predicate Logic, Rules

Artificial Intelligence

Knowledge representation in AI, topics : knowledge progression,

model, category, typology map, and relationship; Mapping between

facts and representation, forward and backward representation, KR

system requirements; KR schemes relational, inheritable, inferential,

declarative and procedural; Issues in KR - attributes, relationship,

granularity. Knowledge representation using logic : Propositional

logic - statements, variables, symbols, connective, truth value,

contingencies, tautologies, contradictions, antecedent, consequent,

argument; Predicate logic predicate, logic expressions, quantifiers,

formula; Representing IsA and Instance relationships;

Computable functions and predicates. Knowledge representation

using rules : declarative, procedural, and meta rules; Logic

programming characteristics - statement, language, syntax, data

objects, clause, predicate, sentence, subject, queries; Programming

paradigms models of computation, imperative, functional, and

logic model; Reasoning, and conflict.

www.myreaders.info


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Artificial Intelligence

Topics(Lectures 15, 16, 17, 18, 19, 20, 21, 22 8 hours)

Slides

1. Knowledge Representation

Introduction Knowledge Progression, KR model, category : typology

map, type, relationship, framework, mapping, forward & backward

representation, KR system requirements; KR schemes - relational,

inheritable, inferential, declarative, procedural; KR issues - attributes,

relationship, granularity.

03-26

2. KR Using Predicate Logic

Logic as language; Logic representation :Propositional logic, statements,

variables, symbols, connective, truth value, contingencies, tautologies,

contradictions, antecedent, consequent, argument; Predicate logic

predicate, logic expressions, quantifiers, formula; Representing IsA

and Instance relationships; Computable functions and predicates;

Resolution.

27-47

3. KR Using Rules

Types of Rules - declarative, procedural, meta rules; Procedural verses

declarative knowledge & language; Logic programming characteristics,

statement, language, syntax & terminology, Data components - simple

& structured data objects, Program Components - clause, predicate,

sentence, subject, queries; Programming paradigms models of

computation, imperative model, functional model, logic model;

Reasoning - Forward and backward chaining, conflict resolution;

Control knowledge.

48-78

4. References 79

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How do we Represent what we know ?

Knowledgeis a general term.An answer to the question, "how to represent knowledge", requires an

analysis to distinguish between knowledge how and knowledge that.

knowing "howto do something".

e.g. "how to drive a car" is a Procedural knowledge.

knowing "thatsomething is true or false".

e.g. "that is the speed limit for a car on a motorway" is a Declarative

knowledge.

knowledge and Representation are two distinct entities. They play a

central but distinguishable roles in intelligent system.

Knowledge is a description of the world.

It determines asystem's competenceby what it knows.

Representation is the way knowledge is encoded.

It defines asystem's performancein doing something.

Different types of knowledge require different kinds of representation.

The Knowledge Representationmodels/mechanisms are often based on:

Logic Rules

Frames Semantic Net

Different types of knowledge require different kinds of reasoning.

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KR -Introduction

1. Introduction

Knowledge is a general term.Knowledge is a progression that starts with datawhich is of limited utility.

By organizing or analyzing the data, we understand what the data means,

and this becomes information.

The interpretation or evaluation of information yield knowledge.

An understanding of the principles embodied within the knowledge is wisdom.

Knowledge Progression

Fig 1 Knowledge Progression

Datais viewed as collection of

disconnected facts.

: Example : It is raining.

Information emerges when

relationshipsamong facts are

established and understood;

Provides answers to "who",

"what", "where", and "when".

: Example : The temperature dropped 15

degrees and then it started raining.

Knowledge emerges when

relationships among patterns

are identified and understood;

Provides answers as "how".

: Example : If the humidity is very high

and the temperature drops substantially,

then atmospheres is unlikely to hold the

moisture, so it rains.

Wisdom is the pinnacle of

understanding, uncovers the

principles of relationships that

describe patterns.

Provides answers as "why".

: Example : Encompasses understanding

of all the interactions that happen

between raining, evaporation, air

currents, temperature gradients andchanges.

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Data Information Knowledge

Organizing

Analyzing

Interpretation

Evaluation

WisdomUnderstanding

Principles


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KR -Introduction

Knowledge Model (Bellinger 1980)

A knowledge model tells, that as the degree of connectedness and

understanding increases, we progress from datathrough information and

knowledgeto wisdom.

Fig. Knowledge Model

The model represents transitionsand understanding.

the transitionsare from data, to information, to knowledge, and finally

to wisdom;

the understanding support the transitions from one stage to the next

stage.

The distinctions between data, information, knowledge, and wisdom are not

very discrete. They are more like shades of gray, rather than black and

white (Shedroff, 2001).

"data" and "information" deal with the past; they are based on the

gathering of factsand adding context.

"knowledge" deals with the presentthat enable us to perform.

"wisdom"deals with the future, acquire vision for what will be, rather

than for what is or was.

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Data

Information

Knowledge

Wisdom

Understandingrelations

Understandingpatterns

Understandingprinciples

Degree ofUnderstanding

Degree of

Connectedness


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KR -Introduction

Knowledge Category

Knowledge is categorized into two major types: Tacit and Explicit.

term Tacitcorresponds to"informal" or "implicit" type of knowledge,

term Explicitcorresponds to "formal" type of knowledge.

Tacit knowledge Explicit knowledge

Exists within a human being;

it is embodied.

Exists outside a human being;

it is embedded.

Difficult to articulate formally.

Can be articulated formally. Difficult to communicate or

share.

Can be shared, copied, processed

and stored.

Hard to steal or copy. Easy to steal or copy

Drawn from experience,

action, subjective insight.

Drawn from artifact of some type as

principle, procedure, process,

concepts.

(The next slide explains more about tacit and explicit knowledge).

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KR -Introduction Knowledge Typology Map

The map shows two types of knowledge - Tacit and Explicitknowledge.

Tacit knowledgecomes from "experience", "action", "subjective" , "insight"

Explicit knowledge comes from "principle", "procedure", "process",

"concepts",via transcribed content or artifact of some type.

Fig. Knowledge Typology Map

Facts : are data or instance that are specific and unique.

Concepts :are class of items, words, or ideas that are known by a

common name and share common features.

Processes: are flow of events or activities that describe how things

work rather than how to do things.

Procedures : are series of step-by-step actions and decisions that

result in the achievement of a task.

Principles :are guidelines, rules, and parameters that govern;

principles allow to make predictions and draw implications;

These artifacts are used in the knowledge creation process to create two

types of knowledge: declarativeand proceduralexplained below.

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Knowledge

TacitKnowledge

Experience

SubjectiveInsight

Doing(action)

ExplicitKnowledge

ProcedurePrinciples

Concept

Process

Information

Context

Data

Facts


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KR -Introduction

Knowledge Type

Cognitive psychologists sort knowledge into Declarative and Procedural

category and some researchers added Strategicas a third category.

Aboutprocedural knowledge, there is some disparity in views.

One, it is close to Tacit knowledge, it manifests itself in the doing of some-thing yet cannot be expressed in words; e.g., we read faces and moods.

Another, it is close to declarative knowledge; the difference is that a task or

method is described instead of facts or things.

All declarative knowledge are explicit knowledge; it is knowledge that

can be and has been articulated.

The strategic knowledge is thought as a subset of declarative

knowledge.

Procedural knowledge Declarative knowledge

Knowledge about "how to dosomething"; e.g., to determine if

Peter or Robert is older, first find

their ages.

Knowledge about "thatsomething is true or false".e.g.,

A car has four tyres; Peter is

older than Robert;

Focuses on tasks that must be

performed to reach a particular

objective or goal.

Refers to representations of

objects and events; knowledge

about facts and relationships;

Examples : procedures, rules,

strategies, agendas, models.

Example : concepts, objects,

facts, propositions, assertions,

semantic nets, logic and

descriptive models.

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KR -Introduction

Relationship among Knowledge Type

The relationship among explicit, implicit, tacit, declarative and procedural

knowledgeare illustrated below.

Fig. Relationship among types of knowledge

The Figure shows :

Declarative knowledge is tied to "describing" and

Procedural knowledge is tied to "doing."

Vertical arrows connecting explicit with declarative and tacit with

procedural, indicate the strong relationships exist among them.

Horizontal arrow connecting declarative and procedural indicates that we

often developprocedural knowledgeas a result of starting with declarative

knowledge. i.e., we often "know about" before we "know how".

Therefore, we may view :

all procedural knowledge as tacit knowledge, and

all declarative knowledge as explicit knowledge.

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Implicit

DoingProceduralDeclarativeDescribing

Explicit Tacit

Has been

articulated

Can not bearticulated

Motor Skill(Manual)

Facts andthings

Tasks andmethods

Mental Skill

KnowledgeStart

No Yes

NoYes


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KR -framework

1.1 Framework of Knowledge Representation (Poole 1998)

Computer requires a well-defined problem description to process and

provide well-defined acceptable solution.

To collect fragments of knowledge we need first to formulate a description

in our spoken language and then represent it in formal language so that

computer can understand. The computer can then use an algorithm to

compute an answer. This process is illustrated below.

Fig. Knowledge Representation Framework

The steps are

Theinformal formalism of the problem takes place first.

It is then represented formally and the computer produces an output.

This output can then be represented in a informally described solution

that user understands or checks for consistency.

Note : The Problem solving requires

formal knowledge representation, and

conversion of informal knowledge to formal knowledge , that is

conversion of implicit knowledge to explicit knowledge.

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Compute

Problem

OutputRepresentation

Solution

Interpret

Solve

Represent

Formal

Informal


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KR - framework

Knowledge and Representation

Problem solving requires large amount of knowledge and some

mechanism for manipulating that knowledge.

The Knowledge and the Representation are distinct entities, play a

central but distinguishable roles in intelligent system.

Knowledge is a description of the world;

it determines a system's competence by what it knows.

Representation is the way knowledge is encoded;

it defines the system's performance in doing something.

In simple words, we:

need to know about things we want to represent, and

need some means by which things we can manipulate.

Objects - facts about objects in the domain.

Events - actions that occur in the domain.

Performance - knowledge about how to do things

know thingsto represent

Meta-knowledge

- knowledge about what we know

need meansto manipulate

Requiressome formalism

- to what we represent ;

Thus, knowledge representation can be considered at two levels :

(a) knowledge level at which facts are described, and

(b) symbol level at which the representations of the objects, defined in

terms of symbols, can be manipulated in the programs.

Note : A good representation enables fast and accurate access to

knowledge and understanding of the content.11


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KR - framework

Mapping between Facts and Representation

Knowledge is a collection of facts from some domain.

We need a representation of "facts" that can be manipulated by a program.

Normal English is insufficient, too hard currently for a computer program to

draw inferences in natural languages.

Thus some symbolic representation is necessary.

Therefore, we must be able to map "facts to symbols" and "symbols to

facts" using forward and backward representation mapping.

Example : Consider an English sentence

Facts Representations

Spot is a dog A factrepresented in English sentence

dog (Spot) Using forward mapping function the

above factis represented in logic

x: dog(x) hastail (x) A logical representationof the factthat"all dogs have tails"

Now using deductive mechanism we can generate a new

representation of object :

hastail (Spot) A new object representation

Spot has a tail

[it is new knowledge]

Using backward mapping function to

generate English sentence

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Facts InternalRepresentation

EnglishRepresentation

Englishunderstanding

English

generation

Reasoning

programs


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KR - framework Forward and Backward Representation

The forward and backward representations are elaborated below :

The doted line on top indicates the abstract reasoningprocess that a

program is intended to model.

The solid lines on bottom indicates the concrete reasoningprocess

that the program performs.

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InitialFacts

InternalRepresentation

EnglishRepresentation

Forwardrepresentationmapping

Backwardrepresentation

mapping

Desired realreasoning Final

Facts

Operated byprogram


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KR - framework

KR System Requirements

A good knowledge representation enables fast and accurate access to

knowledge and understanding of the content.

A knowledge representation system should have following properties.

RepresentationalAdequacy

The ability to represent all kinds of knowledge

that are needed in that domain.

Inferential Adequacy The ability to manipulate the representational

structures to derive new structure corresponding

to new knowledge inferred from old .

Inferential Efficiency The ability to incorporate additional information

into the knowledge structure that can be used tofocus the attention of the inference mechanisms

in the most promising direction.

AcquisitionalEfficiency

The ability to acquire new knowledge using

automatic methods wherever possible rather than

reliance on human intervention.

Note : To date no single system can optimizes all of the above properties.14


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KR - schemes1.2Knowledge Representation Schemes

There are four types of Knowledge representation :

Relational, Inheritable, Inferential, and Declarative/Procedural.

Relational Knowledge :

provides a framework to compare two objects based on equivalentattributes.

any instance in which two different objects are compared is a

relational type of knowledge.

Inheritable Knowledge

is obtained from associated objects.

it prescribes a structure in which new objects are created which may

inherit all or a subset of attributes from existing objects.

Inferential Knowledge

is inferred from objects through relations among objects.

e.g., a word alone is a simple syntax, but with the help of other

words in phrase the reader may infer more from a word; this

inference within linguistic is called semantics.

Declarative Knowledge

a statement in which knowledge is specified, but the use to which

that knowledge is to be put is not given.

e.g. laws, people's name; these are facts which can stand alone, not

dependent on other knowledge;

Procedural Knowledge

a representation in which the control information, to use the

knowledge, is embedded in the knowledge itself.

e.g. computer programs, directions, and recipes; these indicate

specific use or implementation;

These KR schemes are detailed in next few slides15


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KR - schemes

Relational Knowledge :

This knowledge associates elements of one domain with another domain.

Relational knowledge is made up of objects consisting of attributes and

their corresponding associated values.

The results of this knowledge type is a mapping of elements amongdifferent domains.

The table below shows a simple way to store facts.

The facts about a set of objects are put systematically in columns.

This representation provides little opportunity for inference.

Table - Simple Relational Knowledge

Player Height Weight Bats - Throws

Aaron 6-0 180 Right - Right

Mays 5-10 170 Right - Right

Ruth 6-2 215 Left - Left

Williams 6-3 205 Left - Right

Given the facts it is not possible to answer simple question such as :

" Who is the heaviest player ? ".

but if a procedure for finding heaviest player is provided, then these

facts will enable that procedure to compute an answer.

We can ask things like who "bats left" and "throws right".

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KR - schemes[from previous slide example]

Viewing a node as a frame

Example : Baseball-player

isa : Adult-Male

Bates : EQUAL handed

Height : 6.1Batting-average : 0.252

Algorithm : Property Inheritance

Retrieve a value V for an attribute A of an instance object O.

Steps to follow:

1. Find objectOin the knowledge base.

2. If there is a value for the attributeA then report that value.

3. Else, if there is a value for the attribute instance; If not, then fail.

4. Else, move to the node corresponding to that value and look for a

value for the attribute A; If one is found, report it.

5. Else, do until there is no value for theisaattribute or

until an answer is found :

(a) Get the value of theisaattribute and move to that node.

(b) See if there is a value for the attribute A; If yes, report it.

This algorithm is simple. It describes the basic mechanism of

inheritance. It does not say what to do if there is more than one value

of the instance orisaattribute.

This can be applied to the example of knowledge base illustrated, in the

previous slide, to derive answers to the following queries :

team (Pee-Wee-Reese) = BrooklynDodger

battingaverage(Three-Finger-Brown) = 0.106

height (Pee-Wee-Reese) = 6.1

bats (Three Finger Brown) = right

[For explanation - refer book on AI by Elaine Rich & Kevin Knight, page 112]

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KR - schemes

Inferential Knowledge :

This knowledge generates new informationfrom the given information.

This new information does not require further data gathering form source,

but does require analysis of the given information to generate new

knowledge.Example :

given a set of relations and values, one may infer other values or

relations.

a predicate logic (a mathematical deduction) is used to infer from a set

of attributes.

inference through predicate logic uses a set of logical operations to relate

individual data.

the symbols used for the logic operations are :

" " (implication), " " (not), " V " (or), " " (and),

" " (for all), " " (there exists).Examples of predicate logic statements :

1. "Wonder"is a name of a dog : dog (wonder)

2. All dogs belong to the class of animals : x : dog (x) animal(x)3. All animals either live on land or in

water :x : animal(x) live (x,land) V live (x, water)

From these three statements we can infer that :

" Wonder lives either on land or on water."

Note :If more information is made available about these objects and their

relations, then more knowledge can be inferred.

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KR - schemes

Declarative/Procedural Knowledge

Differences between Declarative/Procedural knowledge is not very clear.

Declarative knowledge :

Here, the knowledge is based on declarative facts about axioms and

domains . axioms are assumed to be true unless a counter example is found to

invalidate them.

domains represent the physical world and the perceived functionality.

axiom and domains thus simply exists and serve as declarative

statements that can stand alone.

Procedural knowledge:

Here, the knowledge is a mapping process between domains that specifywhat to do whenand the representation is of how to make it rather

than what it is. The procedural knowledge :

may have inferential efficiency, but no inferential adequacy and

acquisitional efficiency.

are represented as small programs that know how to do specific things,

how to proceed.

Example: A parser in a natural language has the knowledge that a noun

phrase may contain articles, adjectives and nouns. It thus accordingly call

routines that know how to process articles, adjectives and nouns.

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KR - issues

Important Attributes : (Ref. Example - Fig. Inheritable KR)

There are attributesthat are of general significance.

There are two attributes "instance" and "isa", that are of general

importance. These attributes are important because they support

property inheritance.

Relationship among Attributes : (Ref. Example- Fig. Inheritable KR)

The attributes to describe objects are themselves entities they represent.

The relationship between the attributes of an object, independent of specific

knowledge they encode, may hold properties like:

Inverses, existence in an isa hierarchy, techniques for reasoning about

values and single valued attributes.

Inverses :

This is aboutconsistency check,while a value is added to one attribute.

The entities are related to each other in many different ways. The figure

shows attributes (isa, instance, and team), each with a directed arrow,

originating at the object being described and terminating either at the

object or its value.

There are two ways of realizing this:

first, represent two relationships in a single representation; e.g., a

logical representation, team(Pee-Wee-Reese, BrooklynDodgers),

that can be interpreted as a statement about Pee-Wee-Reese or

BrooklynDodger.

second, use attributes that focus on a single entity but use them in

pairs, one the inverse of the other; for e.g., one, team = Brooklyn

Dodgers, and the other, team = Pee-Wee-Reese, . . . .

This second approach is followed in semantic net and frame-based

systems, accompanied by a knowledge acquisition tool that guarantees

the consistency of inverse slot by checking, each time a value is added

to one attribute then the corresponding value is added to the inverse.

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KR - issues

Existence in an "isa" hierarchy:

This is about generalization-specialization, like, classes of objects and

specialized subsets of those classes. There are attributes and

specialization of attributes.

Example: the attribute "height" is a specialization of general attribute

"physical-size"which is, in turn, a specialization of "physical-attribute".

These generalization-specialization relationships for attributes are

important because they support inheritance.

Techniques for reasoning about values:

This is about reasoning values of attributes not given explicitly.

Several kinds of information are used in reasoning, like,

height : must be in a unit of length,

age : of person can not be greater than the age ofperson's parents.

The values are often specified when a knowledge base is created.

Single valued attributes:

This is abouta specificattribute that is guaranteed to take a unique

value.

Example : A baseball player can at time have only a single height and

be a member of only one team. KR systems take different approaches

to provide support for single valued attributes.

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KR - issues

Choosing Granularity

What level should the knowledge be represented and

what are the primitives ?

Should there be a small number or should there be a large number of

low-level primitives or High-level facts.

High-level facts may not be adequate for inference while Low-level

primitives may require a lot of storage.

Example of Granularity:

Suppose we are interested in following facts

John spotted Sue.

This could be represented as

Spotted (agent(John), object (Sue))

Such a representation would make it easy to answer questions such are

Who spotted Sue ?

Suppose we want to know

Did John see Sue ?

Given only one fact, we cannot discover that answer.

We can add other facts, such as

Spotted (x , y) saw (x , y)

We can now infer the answer to the question.

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KR - issues

Set of Objects

Certain properties of objects that are true as member of a set but

not as individual;

Example : Consider the assertion made in the sentences

"there are more sheepthanpeoplein Australia", and

"Englishspeakers can be found all over the world."

To describe these facts, the only way is to attach assertion to the sets

representing people, sheep, and English.

The reason to represent sets of objects is :

If a property is true for all or most elements of a set,

then it is more efficient to associate it once with the set

rather than to associate it explicitly with every elements of the set .

This is done in different ways :

in logical representation through the use of universal quantifier, and

in hierarchical structure where node represent sets, the inheritance

propagateset level assertion down to individual.

Example: assert large (elephant);

Remember to make clear distinction between,

whether we are asserting some property of the set itself,

means, the set of elephants is large, or

asserting some property that holds for individual elements of the set ,means, any thing that is an elephant is large.

There are three ways in which sets may be represented :

(a)Name, as in the example Ref Fig. Inheritable KR, the node - Baseball-

Player and the predicates as Ball and Batter in logical representation.

(b)Extensional definition is to list the numbers, and

(c)In tensional definition is to provide a rule, that returns true or false

depending on whether the object is in the set or not.

[Readers may refer book on AI by Elaine Rich & Kevin Knight- page 122 - 123]

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KR - issues

Finding Right Structure

Access to right structure for describing a particular situation.

It requires, selecting an initial structure and then revising the choice.

While doing so, it is necessary to solve following problems :

how to perform an initial selection of the most appropriate structure.

how to fill in appropriate details from the current situations.

how to find a better structure if the one chosen initially turns out not to

be appropriate.

what to do if none of the available structures is appropriate.

when to create and remember a new structure.

There is no good, general purpose method for solving all these problems.

Some knowledge representation techniques solve some of them.

[Readers may refer book on AI by Elaine Rich & Kevin Knight- page 124 - 126]

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KR using logic

2. KR Using Predicate Logic

In the previous section much has been illustrated about knowledge and KR

related issues. This section, illustrates :

How knowledge can be represented as symbol structures that characterize

bits of knowledge about objects, concepts, facts, rules, strategies;

Examples : red represents colour red;

car1 represents my car;

"red(car1)" represents fact that my car is red.

Assumptions about KR :

Intelligent Behaviorcan be achieved by manipulation of symbol structures.

KR languagesare designed to facilitate operations over symbol structures,

have precise syntax and semantics;Syntax tells which expression is legal ?,

e.g., red1(car1), red1 car1, car1(red1), red1(car1 & car2) ?; and

Semantic tells what an expression means ?

e.g., property dark red applies to my car.

Make Inferences, draw new conclusions from existing facts.

To satisfy these assumptions about KR, we need formal notation that allow

automated inference and problem solving.One popular choice is use oflogic.

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KR Logic

Logic

Logic is concerned with the truth of statements about the world.

Generally each statement is either TRUEor FALSE.

Logic includes : Syntax, Semantics and InferenceProcedure.

Syntax :

Specifies the symbols in the language about how they can be combined

to form sentences. The facts about the world are represented as

sentences in logic.

Semantic :

Specifies how to assign a truth value to a sentence based on its

meaningin the world. It Specifies what facts a sentence refers to.

A fact is a claim about the world, and it may be TRUEor FALSE.

Inference Procedure:

Specifies methods for computing new sentences from the existing

sentences.

Note

Facts : are claims about the world that are Trueor False.

Representation : is an expression (sentence), stands for the objects and

relations.

Sentences : can be encoded in a computer program.

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KR - Logic

Logic as a KR Language

Logic is a language for reasoning, a collection of rules used while doing

logical reasoning. Logic is studied as KR languages in artificial intelligence.

Logicis a formal system in which the formulas or sentences have true

or false values.

Problem of designing KR languageis a tradeoff between that which is

(a) Expressive enough to represent important objects and relations in a

problem domain.

(b)Efficient enough in reasoning and answering questions about

implicit information in a reasonable amount of time.

Logics are of different types : Propositional logic, Predicate logic,

Temporal logic, Modal logic, Description logicetc;

They represent things and allow more or less efficient inference.

Propositional logic and Predicate logic are fundamental to all logic.

Propositional Logic is the study of statements and their connectivity.

Predicate Logic is the study of individuals and their properties.

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KR Logic2.1 Logic Representation

Logic can be used to represent simple facts.

The facts are claims about the world that are Trueor False.

To build a Logic-based representation :

User defines a set of primitivesymbolsand the associatedsemantics.

Logic defines ways of putting symbols together so that user can define

legal sentences in the language that represent TRUEfacts.

Logic defines ways of inferringnew sentencesfrom existing ones.

Sentences - either TRUEor falsebut not both are calledpropositions.

A declarative sentence expresses a statement with a proposition as

content; example:

the declarative "snow is white"expresses that snow is white;

further, "snow is white"expresses that snow is whiteis TRUE.

In this section, first Propositional Logic (PL) is briefly explained and then

the Predicate logicis illustrated in detail.

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KR - Propositional Logic

Propositional Logic (PL)

A proposition is a statement, which in English would be a declarative

sentence. Every proposition is either TRUE or FALSE.

Examples: (a) The sky is blue., (b) Snow is cold. , (c) 12 * 12=144

Propositions are sentences , either true or false but not both.

A sentence is smallest unit in propositional logic.

If proposition is true, then truth value is "true" .

If proposition is false, then truth value is "false" .

Example :

Sentence Truth value Proposition (Y/N)

"Grass is green" "true" Yes

"2 + 5 = 5" "false" Yes

"Close the door" - No

"Is it hot out side ?" - No

"x > 2" where x is variable - No(since x is not defined)

"x = x" - No

(don't know what is "x" and "=";"3 = 3" or "air is equal to air" or

"Water is equal to water"has no meaning)

Propositional logic is fundamental to all logic.

Propositional logic is also called Propositional calculus, Sentential

calculus, or Boolean algebra.

Propositional logic tells the ways of joining and/or modifying entire

propositions, statements or sentences to form more complicated

propositions, statements or sentences, as well as the logical

relationships and properties that are derived from the methods of

combining or altering statements.

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KR - Propositional Logic Statement, Variables and Symbols

These and few more related terms, such as, connective, truth value,

contingencies, tautologies, contradictions, antecedent, consequent,

argument are explained below.

Statement

Simple statements (sentences), TRUE or FALSE, that does not

contain any other statement as a part, are basic propositions;

lower-case letters, p, q, r, are symbols for simple statements.

Large, compound or complex statement are constructed from basic

propositions by combining them with connectives.

Connective or Operator

The connectives joinsimple statements into compounds, and joins

compounds into larger compounds.

Table below indicates, thebasic connectives and their symbols :

listed in decreasing order of operation priority;

operations with higher priority is solved first.

Example of a formula : ((((a b) V c d) (a V c ))Connectives and Symbols in decreasing order of operation priority

Connective Symbols Read as

assertion P "p is true"

negation p ~ ! NOT "p is false"

conjunction p q && & AND "both p and q are true"disjunction P v q || | OR "either p is true, or q is true, or both "

implication p q if ..then "if p is true, then q is true"" p implies q "

equivalence if and only if "p and q are either both true or both false"

Note : The propositions and connectives are the basic elements of

propositional logic.

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Truth Value

The truth value of a statement is its TRUTHor FALSITY,

Example :

p is eitherTRUEor FALSE,

~p is either TRUEor FALSE,

p v q is either TRUEor FALSE, and so on.

use "T" or "1" to mean TRUE.

use "F" or "0" to mean FALSE

Truth table defining the basicconnectives:

p q p q p q p v q pq p q qpT T F F T T T T T

T F F T F T F F T

F T T F F T T F F

F F T T F F T T T

[The next slide shows the truth values of a group of propositions, called

tautology, contradiction, contingency, antecedent, consequent. They

form argument where one proposition claims to follow logically other

proposition]

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Tautologies

A proposition that is always true is called a "tautology".

e.g., (P v P)is always true regardless of the truth value of theproposition P.

Contradictions

A proposition that is always false is called a "contradiction".

e.g., (P P) is always false regardless of the truth value ofthe proposition P.

Contingencies

A proposition is called a "contingency" , if that proposition is

neither a tautologynor a contradiction.

e.g., (P v Q) is a contingency.

Antecedent, Consequent

These two are parts of conditional statements.

In the conditional statements, p q, the

1st statement or"if - clause"(here p) is called antecedent,

2nd statement or"then - clause"(here q) is called consequent.

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Argument

An argument is a demonstration or a proof of some statement.

Example : "That bird is a crow; therefore, it's black."

Any argumentcan be expressed as a compound statement.

In logic, an argument is a set of one or more meaningful declarative

sentences (or "propositions") known as the premises along with

another meaningful declarative sentence (or "proposition") known

as the conclusion.

Premiseis a proposition which gives reasons, grounds, or evidence for

accepting some other proposition, called the conclusion.

Conclusion is a proposition, which is purported to be established on

the basis of other propositions.

Take all the premises, conjoin them, and make that conjunction theantecedent of a conditional and make the conclusion the

consequent. This implication statement is called the corresponding

conditional of the argument.

Note : Every argument has a corresponding conditional, and every

implication statement has a corresponding argument. Because the

corresponding conditional of an argument is a statement, it is therefore

either a tautology, or a contradiction, or a contingency.

An argument is valid

"if and only if" its corresponding conditional is a tautology.

Two statements are consistent

"if and only if" their conjunction is not a contradiction.

Two statements are logically equivalent

"if and only if"their truth table columns are identical;

"if and only if"the statement of their equivalence using " " isa tautology.

Note : The truth tables are adequate to test validity, tautology,

contradiction, contingency, consistency, and equivalence.

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KR - Predicate Logic

Predicate Logic

The propositional logic, is not powerful enough for all types of assertions;

Example : The assertion "x > 1", where xis a variable, is not a proposition

because it is neither true nor false unless value of xis defined.

For x > 1 to be a proposition ,

either we substitute a specific number for x;

or change it to something like

"There is a number x for which x > 1 holds";

or "For every number x, x > 1 holds".

Consider example :

All men are mortal.

Socrates is a man.

Then Socrates is mortal ,

These cannot be expressed in propositional logic as a finite and logically

valid argument (formula).

We need languages : that allow us to describe properties ( predicates) of

objects, or a relationship among objects represented by the variables .

Predicate logic satisfies the requirements of a language.

Predicate logic is powerful enough for expression and reasoning.

Predicate logic is built upon the ideas ofpropositional logic.

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KR - Predicate Logic Predicate :

Every complete "sentence" contains two parts : a "subject" and a

"predicate".

The subject is what (or whom) the sentence is about.

The predicate tells something about the subject;

Example :

A sentence "Judy{runs}".

The subject isJudy and the predicate is runs.

Predicate, always includes verb, tells something about the subject.

Predicate is a verb phrase template that describes a property of

objects, or a relation among objects represented by the variables.

Example:

The car Tom is driving is blue" ;

"The sky is blue" ;

"The cover of this book is blue"

Predicate is is blue", describes property.

Predicates are given names; Let B is name for predicate "is_blue".

Sentence is represented as "B(x)", read as "x is blue";

Symbol x represents an arbitraryObject .

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KR - Predicate Logic Predicate Logic Expressions:

The propositional operators combine predicates, like

If ( p(....) &&( !q(....) ||r (....) ) )

Logic operators :

Examples of disjunction(OR) and conjunction(AND).Consider the expression with the respective logic symbols ||and &&

x < y ||( y < z &&z < x)

which is true ||( true && true) ;

Applying truth table, found True

Assignment for


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KR - Predicate Logic Predicate Logic Quantifiers

As said before, x > 1 is not proposition and why ?

Also said, that for x > 1 to be a proposition what is required ?

Generally, a predicate with variables (is called atomic formula) that

can be made a proposition by applying one of the following two

operations to each of its variables :

1. Assign a value to the variable; e.g., x > 1, if 3 is assigned tox

becomes 3 > 1 , and it then becomes a true statement, hence a

proposition.

2. Quantify the variable using a quantifier on formulas of predicate

logic (called wff well-formed formula), such as x > 1or P(x), by

using Quantifiers on variables.

Apply Quantifiers on Variables

Variable x

* x > 5 is not a proposition, its truth depends upon the

value of variablex

* to reason such statements,x need to be declared

Declaration x : a

* x : a declares variable x* x : a read as x is an element of set a

Statement p is a statement about x

* Q x : a p is quantification of statement

statement

declaration of variable xas element of set a

quantifier

* Quantifiers are two types :

universal quantifiers , denoted by symbol and

existential quantifiers , denoted by symbol

Note : The next few slide tells more on these two Quantifiers.

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KR - Predicate Logic Apply Universal Quantifier " For All "

Universal Quantification allows us to make a statement about a

collection of objects.

Universal quantification: x : a p

* read for all x in a, p holds

* a is universe of discourse

* x is a member of the domain of discourse.

* p is a statement about x

In propositional form it is written as : x P(x)

* read for all x, P(x)holds

for each x, P(x) holds or

for every x, P(x) holds

* where P(x) is predicate,

x means all the objects x in the universe

P(x) is true for every object x in the universe

Example : English language to Propositional form

* "All cars have wheels"

x : car x has wheel

* x P(x)

where P (x)is predicate tells : x has wheels

x is variable for object cars that populateuniverse of discourse

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Apply Existential Quantifier " There Exists "

Existential Quantification allows us to state that an object does exist

without naming it.

Existential quantification: x : a p

* read there exists an xsuch that p holds

* a is universe of discourse

* x is a member of the domain of discourse.

* p is a statement about x

In propositional form it is written as : x P(x)

* read there exists an xsuch that P(x) or

there exists at least one xsuch that P(x)

* Where P(x) is predicate

x means at least one object x in the universe

P(x) is true for least one object x in the universe

Example : English language to Propositional form

* Someone loves you

x : Someone x loves you

* x P(x)

where P(x) is predicate tells : x loves you

x is variable for object someone that populateuniverse of discourse

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Formula

In mathematical logic, a formula is a type of abstract object.

A token of a formula is a symbol or string of symbols which may be

interpreted as any meaningful unit in a formal language.

Terms

Defined recursively as variables, or constants, or functions like

f(t1, . . . , tn), where fis an n-ary function symbol, and t1, . . . , tn

are terms. Applying predicates to terms produces atomic formulas.

Atomic formulas

An atomic formula (or simply atom) is a formula with no deeper

propositional structure, i.e., a formula that contains no logical

connectives or a formula that has no strict sub-formulas.

Atomsare thus the simplest well-formed formulas of the logic.

Compound formulas are formed by combining the atomic

formulas using the logical connectives.

Well-formed formula("wiff") is a symbol or string of symbols (a

formula) generated by the formal grammar of a formal language.

An atomic formula is one of the form :

t1= t2, where t1 and t2 are terms, or

R(t1, . . . , tn), where R is an n-ary relation symbol, and

t1, . . . , tn are terms.

a is a formula when a is a formula.

(a b)and (a v b) are formula when a and bare formula Compound formula : example

((((a b ) c) (( a b) c)) ((a b) c))43


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KR logic relation2.2 Representing IsA and Instance Relationships

Logic statements, containing subject, predicate, and object, were

explained. Also stated, two important attributes "instance" and "isa", in

a hierarchical structure (Ref. Fig. Inheritable KR).

Attributes IsA and Instance support property inheritance and play

important role in knowledge representation.

The ways these two attributes "instance"and "isa", are logically expressed

are shown in the example below :

Example : A simple sentence like "Joe is a musician"

Here "is a" (called IsA) is a way of expressing what logically

is called a class-instance relationship between the subjects

represented by the terms "Joe" and "musician".

"Joe" is an instance of the class of thingscalled "musician".

"Joe" plays the role of instance,

"musician" plays the role of classin that sentence.

Note : In such a sentence, while for a human there is no confusion,

but for computers each relationship have to be defined explicitly.

This is specified as: [Joe] IsA [Musician]

i.e., [Instance] IsA [Class]

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KR functions & predicates

2.3 Computable Functions and Predicates

The objective is to define class of functions C computable in terms of F.

This is expressed as C { F } is explained below using two examples :

(1) "evaluate factorial n" and (2) "expression for triangular functions".

Example 1 : A conditional expression to define factorial n ie n!

Expression

if p1then e1 else if p2then e2 . . . else if pnthen en.

ie. (p1 e1, p2 e2, . . . . . . pn en)

Here p1, p2, . . . . pn are propositional expressions taking the

values Tor F for true and false respectively.

The value of( p1 e1, p2 e2, . . . . . .pn en ) is the value of the

ecorresponding to the first pthat has value T. The expressions defining n! , n= 5, recursively are :

n! = n x (n-1)! for n 15! = 1 x 2 x 3 x 4 x 5 = 120

0! = 1

The above definition incorporates aninstance that :

if the product of no numbers ie 0! = 1,

then only, recursive relation(n + 1)! = (n+1) x n! works forn = 0

Use of the above conditional expressions to define functions n!

recursively is n! = ( n = 0 1, n 0 n . (n 1 ) ! )

Example: Evaluate 2! according to above definition.

2! = ( 2 = 0 1, 2 0 2 . ( 2 1 )! )

= 2 x 1!

= 2 x ( 1 = 0 1, 1 0 1 . ( 1 1 )! )

= 2 x 1 x 0!

= 2 x 1 x ( 0 = 0 1, 0 0 0 . ( 0 1 )! )

= 2 x 1 x 1

= 2

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KR functions & predicates Example 2 : A conditional expression for triangular functions

The graph of a well known triangular function is shown below

Fig. A Triangular Function

the conditional expressions for triangular functions are

x= (x 1 0)46

Y

-1,0 1,0X

0,1


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KR - Predicate Logic resolution2.4 Resolution

Resolution is a procedure used in proving that arguments which are

expressible in predicate logic are correct.

Resolution is a procedure that produces proofs by refutation or

contradiction.

Resolution lead to refute a theorem-proving technique for sentences in

propositional logic and first-order logic.

Resolution is a rule of inference.

Resolution is a computerized theorem prover.

Resolution is so far only defined for Propositional Logic. The strategy is

that the Resolution techniques of Propositional logic be adopted in

Predicate Logic.

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KR Using Rules

3. KR Using Rules

In the earlier slides, the Knowledge representations using predicate logic

have been illustrated. The other popular approaches to Knowledge

representation are called production rules, semantic net and frames.

Production rules, sometimes called IF-THENrules are most popular KR.

production rules are simple but powerful forms of KR.

production rules provide the flexibility of combining declarative and

procedural representation for using them in a unified form.

Examples of production rules :

IF condition THEN action

IF premise THEN conclusion

IF proposition p1and proposition p2are true THEN proposition p3is true

Advantages of production rules :

they are modular,

each rule define a small and independent piece of knowledge.

new rules may be added and old ones deleted

rules are usually independently of other rules.

The production rules as knowledge representation mechanism are used in the

design of many"Rule-based systems" also called "Production systems".

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KR Using Rules

Types of Rules

Three types of rules are mostly used in the Rule-based production systems. Knowledge Declarative Rules :

These rules state all the facts and relationships about a problem.

Example :

IF inflation rate declines

THEN the price of gold goes down.

These rules are a part of the knowledge base.

Inference Procedural Rules

These rules advise on how to solve a problem, while certain facts are

known.

Example :

IF the data needed is not in the system

THEN request it from the user.

These rules are part of the inference engine.

Meta rules

These are rules for making rules. Meta-rules reason about which rules

should be considered for firing.

Example :

IF the rules which do not mention the current goal in their premise,

AND there are rules which do mention the current goal in their premise,

THEN the former rule should be used in preference to the latter.

Meta-rules direct reasoning rather than actually performing

reasoning.

Meta-rules specify which rules should be considered and in which

order they should be invoked.

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KR procedural & declarative

3.1 Procedural versus Declarative Knowledge

These two types of knowledge were defined in earlier slides.

Procedural Knowledge: knowing 'how to do'

Includes : rules, strategies, agendas, procedures, models.

These explains what to doin order to reach a certain conclusion.Example

Rule: To determine if Peter or Robert is older, first find their ages.

It is knowledge about 'how to do' something. It manifests itself in the

doing of something, e.g., manual or mental skills cannot reduce to

words. It is held by individuals in a way which does not allow it to be

communicated directly to other individuals.

Accepts a description of the steps of a task or procedure. It Looks

similar to declarative knowledge, except that tasks or methods are

being described instead of facts or things.

Declarative Knowledge: knowing 'what', knowing 'that'

Includes : concepts, objects, facts, propositions, assertions, models.

It is knowledge about factsand relationships, that

can be expressed in simple and clear statements,

can be added and modified without difficulty.Examples : A car has four tyres; Peter is older than Robert.

Declarative knowledge and explicit knowledge are articulated knowledge

and may be treated as synonyms for most practical purposes.

Declarative knowledge is represented in a format that can be

manipulated, decomposed and analyzed independent of its content.

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KR procedural & declarative Comparison :

Comparison between Procedural and Declarative Knowledge :

Procedural Knowledge Declarative Knowledge

Hard to debug Easy to validate

Black box White box

Obscure Explicit

Process oriented Data - oriented

Extension may effect stability Extension is easy

Fast , direct execution Slow (requires interpretation)

Simple data type can be used May require high level data type

Representations in the form ofsets of rules, organized into

routines and subroutines.

Representations in the form ofproduction system, the entire set

of rules for executing the task.

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KR procedural & declarative Comparison :

Comparison between Procedural and Declarative Language :

Procedural Language Declarative Language

Basic, C++, Cobol, etc. SQL

Most work is done by interpreter of

the languages

Most work done by Data Engine

within the DBMS

For one task many lines of code For one task one SQL statement

Programmer must be skilled intranslating the objective into linesof procedural code

Programmer must be skilled inclearly stating the objective as aSQL statement

Requires minimum of managementaround the actual data

Relies on SQL-enabled DBMS tohold the data and execute the SQLstatement .

Programmer understands and hasaccess to each step of the code

Programmer has no interactionwith the execution of the SQLstatement

Data exposed to programmerduring execution of the code

Programmer receives data at endas an entire set

More susceptible to failure due tochanges in the data structure

More resistant to changes in thedata structure

Traditionally faster, but that ischanging

Originally slower, but now settingspeed records

Code of procedure tightly linked tofront end

Same SQL statements will workwith most front endsCode loosely linked to front end.

Code tightly integrated withstructure of the data store

Code loosely linked to structure ofdata; DBMS handles structuralissues

Programmer works with a pointeror cursor

Programmer not concerned withpositioning

Knowledge of coding tricks appliesonly to one language Knowledge of SQL tricks applies toany language using SQL

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KR Logic Programming

3.2 Logic Programming

Logic programming offers a formalism for specifying a computation in terms

of logical relations between entities.

logic programis a collection of logic statements.

programmer describes all relevant logical relationships between the

various entities.

computationdetermines whether or not, a particular conclusion follows

from those logical statements.

Characteristics of Logic program

Logic program is characterized by set of relations and inferences.

programconsists of a set of axioms and a goal statement.

rules of inferencedetermine whether the axioms are sufficient to ensurethe truth of the goal statement.

executionof a logic program corresponds to the construction of a proof

of the goal statement from the axioms.

programmer specify basic logical relationships, does not specify the

manner in which inference rules are applied.

Thus Logic + Control = Algorithms

Examples of Logic Statements

Statement

A grand-parent is a parent of a parent.

Statement expressed in more closely related logic terms as

A person is a grand-parent if she/he has a child and

that child is a parent.

Statement expressed in first order logic as

(for all) x: grandparent (x, y) :- parent (x, z), parent (z, y)

read as x is the grandparent of yif x is a parent of z and z is a parent of y

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Logic Programming Language

A programming language includes :

the syntax

the semanticsof programs and

the computational model.

There are many ways of organizing computations. The most familiar

paradigm is procedural. The program specifies a computation by saying

"how" it is to be performed. FORTRAN, C,and Object-oriented languages

fall under this general approach.

Another paradigm is declarative. The program specifies a computation by

giving the properties of a correct answer. Prolog and logic data language

(LDL) are examples of declarative languages, emphasize the logicalproperties of a computation.

Prolog and LDL are called logic programming languages.

PROLOG (PROgramming LOGic) is the most popular Logic programming

language rose within the realm of Artificial Intelligence (AI). It became

popular with AI researchers, who know more about "what" and "how"

intelligent behavior is achieved.

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Syntax and Terminology (relevant to Prolog programs)

In any language, the formation of components (expressions, statements,

etc.), is guided by syntactic rules.

The components are divided into two parts:

(A) data components and (B) program components.

(A) Data components:

Data components are collection of data objects that follow hierarchy.

Data object of any kind is also called

a term. A term is a constant, a variable

or a compound term.

Simple data object is not

decomposable; e.g. atoms, numbers,

constants, variables.

Syntaxdistinguishes the data objects,

hence no need for declaring them.

Structured data object are made of

several components.

All these data components are explained in next slide.

55

Data Objects(terms)

Simple Structured

VariablesConstants

NumbersAtoms


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KR Logic Programming(a) Data Objects :

The data objects of any kind is called a term.

Term: Examples

Constants:

Denote elements such as integers, floating point, atoms.

Variables:

Denote a single but unspecified element; symbols for variables

begin with an uppercase letter or an underscore.

Compound terms:

Comprise a functor and sequence of one or more compound

terms called arguments.

Functor: is characterized by its name and number ofarguments; name is an atom, and number of arguments is

arity.

/n = ( t 1, t2, . . . tn)

where is name of the functorand is of arityn

ti's are the argument

/n denotes functorof arity n

Functors with same name but different arities are distinct.

Ground and non-ground:

Terms are ground if they contain no variables (only constant

signs); otherwise they are non-ground.

Goals are atoms or compound terms, and are generally non-

ground.

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KR Logic Programming(b)Simple Data Objects :Atoms, Numbers, Variables

Atoms

a lower-case letter, possibly followed by other letters of either

case, digits, and underscore character.

e.g. a greaterThan two_B_or_not_2_b

a string of special characters such as:+ - * / \ = < > : ~ # $ &

e.g. ##&& ::=

a string of any characters enclosed within single quotes.

e.g. 'ABC' '1234' 'ab'

following are also atoms ! ; [] {}

Numbers

applications involving heavy numerical calculations are rarely

written in Prolog.

integer representation: e.g. 0 -16 33 +100

real numbers written in standard or scientific notation,

e.g. 0.5 -3.1416 6.23e+23 11.0e-3 -2.6e-2

Variables

begins by a capital letter, possibly followed by other letters of

either case, digits, and underscore character.

e.g. X25 List Noun_Phrase

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KR Logic Programming(c) Structured Data Objects: General Structures , Special Structures

General Structures

a structured term is syntactically formed by a functor and a list of

arguments.

functor is an atom.

list of arguments appear between parentheses.

arguments are separated by a comma.

each argument is a term (i.e., any Prolog data object).

the number of arguments of a structured term is called its arity.

e.g. greaterThan(9, 6) f(a, g(b, c), h(d)) plus(2, 3, 5)

Note : a structure in Prolog is a mechanism for combining termstogether, like integers 2, 3, 5 are combined with the functor plus.

Special Structures

In Prolog an ordered collection of terms is called a list.

Lists are structured terms and Prolog offers a convenient

notation to represent them:

* Empty list is denoted by the atom[ ].

* Non-empty list carries element(s) between square brackets,

separating elements by comma.

e.g. [bach, bee] [apples, oranges, grapes]

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KR Logic Programming(B)Program Components

A Prolog program is a collection of predicates or rules.

A predicate establishes a relationships between objects.

(a) Clause, Predicate, Sentence, Subject

Clause is a collection of grammatically-related words .

Predicate is composed of one or more clauses.

Clauses are the building blocks of sentences;

every sentence contains one or more clauses.

A Complete Sentence has two parts: subject and predicate.

o subject is what (or whom) the sentence is about.

o predicate tells something about the subject.

Example 1 : "cows eat grass".

It is a clause, because it contains

the subject "cows" and

the predicate "eat grass."

Example 2 : "cows eating grass are visible from highway"

This is a complete clause.

the subject "cows eating grass" andthe predicate "are visible from the highway" makes complete

thought.

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KR Logic Programming(b)Predicates & Clause

Syntactically a predicate is composed of one or more clauses.

The general form of clauses is

:- .

where LHS is a single goal called "goal" and RHS is composed ofone or more goals, separated by commas, called "sub-goals"of

the goal on left-hand side.

Thesymbol " :-" is pronouncedas "it is the case" or "such that"

The structure of a clause in logic programhead body

pred( functor(var1, var2)) :- pred(var1) , pred(var2)

literal literal

clause

Literals represent the possible choices in primitive types the

particular language. Some of the choices of types of literals are

often integers, floating point, Booleans and character strings.

Example : grand_parent (X, Z) :- parent(X, Y), parent(Y, Z).

parent (X, Y) :- mother(X, Y).

parent (X, Y) :- father(X, Y).

Read as if x is mother of y then x is parent of y

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KR Logic Programming[Continued from previous slide]

Interpretation:

* A clause specifies the conditional truth of the goal on the LHS;

goal on LHS is assumed to be true if the sub-goals on RHS are all

true. A predicate is true if at least one of its clauses is true.

* An individual "X" is thegrand-parent of "Z" if aparentof that

same"X" is "Y" and "Y" is the parent of that "Z".

X Y Z

* Anindividual "X" is a parent of "Y" if "Y"is the mother of "X"

X Y

* Anindividual "X" is a parent of "Y" if "Y" is the father of "X".

X Y

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(Y is parent of Z)(X is parent of Y)

(X is grand parent of Z)

(X is parent of Y)

(X is mother of Y)

(X is parent of Y)

(X is father of Y)


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KR Logic Programming(c)Unit Clause- a special Case

Unlike the previous example of conditional truth, one often encounters

unconditional relationships that hold.

In Prolog the clauses that are unconditionally true are called

unit clause or fact .

Example : Unconditionally relationships say

'X' is the father of 'Y' is unconditionally true.

This relationship as a Prolog clauseis

father(X, Y) :- true.

Interpreted as relationship of father betweenX andY is always

true; or simply stated as X is father of Y.

Goal true is built-in in Prolog and always holds.

Prolog offers a simpler syntax to express unit clause or fact

father(X, Y)

ie the ":- true" part is simply omitted.

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KR Logic Programming(d) Queries

In Prolog the queries are statements called directive.

A special case of directives, are called queries.

Syntactically, directives are clauses with an empty left-hand side.

Example : ? - grandparent(Q,Z).

This query Qis interpreted as : Who is a grandparent of Z?

By issuing queries Q, Prolog tries to establish the validity of specific

relationships.

The answer from previous slides is (Xis grand parent of Z)

The result of executing a query is either success or failure

Success, means the goals specified in the query holds according to

the facts and rules of the program.

Failure, means the goals specified in the query does not hold

according to the facts and rules of the program.

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KR Logic - models of computation

Programming Paradigms : Models of Computation

A complete description of a programming language includes the

computational model, syntax, semantics, and pragmatic considerations

that shape the language.

Models of Computation:

A computational model is a collection of values and operations, while

computation is the application of a sequence of operations to a value

to yield another value.

There are three basic computational models :

(a) Imperative, (b) Functional, and (c) Logic.

In addition to these, there are two programming paradigms :

(a) concurrent (b) object-oriented programming .

While, these two are not models of computation, but they rank in

importance with computational models.

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KR Logic - models of computation(a)Imperative Model

The Imperative model of computation, consists of a state and an

operation of assignment which is used to modify the state.

Programs consist of sequences of commands.

Computations are changes in the state.

Example : Linear function

A linear function y = 2x + 3 can be written as

Y := 2 X + 3The implementation requires to determines the value of X in the state

and then creates a new state which differs from the old state.

New State: X = 3, Y = 9,

The imperative model is closest to the hardware model on whichprograms are executed, that makes it most efficient model in terms

of execution time.

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KR Logic - models of computation(c)Logic Model

The logic model of computation is based on relations and logical

inference.

Programs consist of definitions of relations.

Computations are inferences (is a proof).

Example 1: Linear function

A linear function y = 2x + 3 can be represented as :

f (X , Y) if Y is 2 X + 3.Here the function represents the relation between Xand Y.

Example 2: Determine a value for Circumference.

The circumference computation can be represented as:

Circle (R , C) if Pi = 3.14 and C = 2 pi R.Here the function is represented as the relation between radius R

and circumference C.

Example 3: Determine the mortality of Socrates and Penelope.

The program is to determine the mortality of Socrates and Penelope.

The fact given that Socrates and Penelope are human.

The rule is that all humans are mortal, that is

for all X, if X is human then X is mortal.

To determine the mortality of Socrates or Penelope, make the

assumption that there are no mortals, that is mortal (Y)

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KR Logic - models of computation[logic model continued in the previous slide]

The equivalent form of the facts and rules stated before are

human (Socrates)

mortal (X) if human (X)

To determine the mortality of Socrates and Penelope, we made the

assumption that there are no mortals i.e. mortal (Y) Computation (proof) that Socrates is mortal

1. (a) human(Socrates) Fact

2. mortal(X) if human(X) Rule

3 mortal(Y) assumption

4.(a) X = Y

4.(b) human(Y)from 2 & 3 by unification

and modus tollens

5. Y = Socrates from 1 and 4 by unification

6. Contradiction 5, 4b, and 1

Explanation :

* The 1st line is the statement "Socrates is a man."

* The 2nd line is a phrase "all human are mortal"

into the equivalent "for all X, if X is a man then X is mortal".

* The 3rd line is added to the set to determine the mortality of Socrates.

* The 4th line is the deduction from lines 2 and 3. It is justified by the

inference rule modus tollens which states that if the conclusion of a

rule is known to be false, then so is the hypothesis.

* Variables X and Y are unified because they have same value.

* By unification, Lines 5, 4b, and 1 produce contradictions and identify

Socrates as mortal.

* Note that, resolution is an inference rule which looks for a contradiction

and it is facilitated by unification which determines if there is a

substitution which makes two terms the same.

Logic model formalizes the reasoning process. It is related to relational

data bases and expert systems.

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KR forward-backward reasoning Example 1

Rule R1 : IF hot AND smoky THEN fire

Rule R2 : IF alarm_beeps THEN smoky

Rule R3 : IF fire THEN switch_on_sprinklers

Fact F1 : alarm_beeps [Given]

Fact F2 : hot [Given]

Example 2

Rule R1 : IF hot AND smoky THEN ADD fire

Rule R2 : IF alarm_beeps THEN ADD smoky

Rule R3 : IF fire THEN ADD switch_on_sprinklers



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KR forward-backward reasoning Example 3 : A typical Forward Chaining



Rule R3 : If fire THEN ADD switch_on_sprinklers



Fact F4 : smoky [from F1 by R2]

Fact F2 : fire [from F2, F4 by R1]

Fact F6 : switch_on_sprinklers [from F2 by R3]

Example 4 : A typical Backward ChainingRule R1 : IF hot AND smoky THEN fire

Rule R2 : IF alarm_beeps THEN smoky

Rule R3 : If _fire THEN switch_on_sprinklers



Goal : Should I switch sprinklers on?

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KR forward chaining

Forward Chaining

The Forward chaining system, properties , algorithms, and conflict

resolution strategy are illustrated.

Forward chainingsystem

facts are held in a working memory condition-action rules represent actions to be taken when specified

facts occur in working memory.

typically, actions involve adding or deleting facts from the working

memory.

Properties of Forward Chaining

all rules which can fire do fire.

can be inefficient - lead to spurious rules firing, unfocused problemsolving

set of rules that can fire known as conflict set.

decision about which rule to fire is conflict resolution.

72

rules

WorkingMemory

InferenceEngine

RuleBase

User

facts

facts

facts


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KR forward chaining Forward chainingalgorithm - I

Repeat

Collect the rule whose condition matches a fact in WM.

Do actions indicated by the rule.

(add facts to WM or delete facts from WM)

Until problem is solved or no condition match

Apply on the Example 2 extended (adding 2 more rules and 1 fact)



Rule R3 : If fire THEN ADD switch_on_sprinklers

Rule R4 : IF dry THEN ADD switch_on_humidifier

Rule R5 : IF sprinklers_on THEN DELETE dry



Fact F2 : Dry [Given]

Now, two rules can fire (R2 and R4)

Rule R4 ADD humidifier is on [from F2]

ADD smoky [from F1]

ADD fire [from F2 by R1]

ADD switch_on_sprinklers [by R3]

Rule R2

[followed by

sequence ofactions]

DELEATE dry, iehumidifier is off a conflict !

[by R5 ]

Forward chainingalgorithm - II (applied to example 2 above )

Repeat

Collect the rules whose conditions match facts in WM.

If more than one rule matches as stated above then

Use conflict resolution strategyto eliminate all but one

Do actions indicated by the rules

(add facts to WM or delete facts from WM)

Until problem is solved or no condition match

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KR forward chaining Alternative to Conflict Resolution Use Meta Knowledge

Instead of conflict resolution strategies, sometimes we want to use

knowledge in deciding which rules to fire. Meta-rules reason about

which rules should be considered for firing. They direct reasoning rather

than actually performing reasoning.

Meta-knowledge : knowledge about knowledge to guide search.

Example of meta-knowledge

IF conflict set contains any rule (c , a) such that

a = "animal is mammal''

THEN fire (c , a)

This example says meta-knowledge encodes knowledge about how

to guide search for solution.

Meta-knowledge, explicitly coded in the form of rules with "object

level" knowledge.

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KR backward chaining

Backward Chaining

Backward chaining system and the algorithm are illustrated.

Backward chainingsystem

Backward chaining means reasoning from goals back to facts.

The idea is to focus on the search.

Rules and

03 knowledge representations

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